Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,23,20)(19,21,24,26)(22,25,30,28)(27,29,31,32)(33,34,39,38)(35,40,45,44)(36,46,47,41)(37,42,43,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,35,39,45)(34,40,38,44)(36,37,47,43)(41,48,46,42);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,36,39,47)(34,41,38,46)(35,43,45,37)(40,42,44,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,37)(34,42)(35,36)(38,48)(39,43)(40,46)(41,44)(45,47);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,44)(43,45);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,38,39,34)(35,44,45,40)(36,46,47,41)(37,42,43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=hi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8488637);